A Characterization of Generalized Zeros of Negative Type of Functions of the Class Nκ View Full Text


Ontology type: schema:Chapter     


Chapter Info

DATE

1986

AUTHORS

Heinz Langer

ABSTRACT

Recall ([1], [2], [3]) that Nκ denotes the set of all complex valued functions Q which are meromorphic in the open upper half plane C + and such that the kernel NQ: $${N_Q}\left( {z,\zeta } \right):\left( {Q\left( z \right) - \overline {Q\left( \zeta \right)} } \right)/\left( {z - \overline \zeta } \right)$$ (1.1) for z,ζ ε D Q has κ negative squares (here D Q (⊂C +) denotes the domain of holomorphy of Q). This means that for arbitrary n ε Z and z1,z2,...,zn ε D Q the matrix (NQ(zi,zj)) 1 n has at most κ negative eigenvalues and for at least one choice of n, z1,...,zn it has exactly κ negative eigenvalues. The class No coincides with the Nevanlinna class of all functions which are holomorphic in C + and map C + into C + UR. The following two examples of functions of the class N1 were considered in [2], [4], respectively: $$w\left( z \right):\alpha - z + \int\limits_{ - \infty }^\infty {\left( {{{\left( {t - z} \right)}^{ - 1}} - t{{\left( {1 + {t^2}} \right)}^{ - 1}}} \right)} d{\sigma _O}\left( t \right),v\left( z \right): = \alpha + \left( {1/z} \right) + \int\limits_{ - 8}^\infty {\left( {{{\left( {t - z} \right)}^{ - 1}} - t{{\left( {1 + {t^2}} \right)}^{ - 1}}} \right)} d{\sigma _1}\left( t \right),$$ (1.2) where α ε R and σo, σl are nondecreasing functions on R such that $${\int\limits_{ - \infty }^\infty {\left( {1 + {t^2}} \right)} ^{ - 1}}d{\sigma _j}\left( t \right) More... »

PAGES

201-212

Book

TITLE

Advances in Invariant Subspaces and Other Results of Operator Theory

ISBN

978-3-0348-7700-8
978-3-0348-7698-8

Author Affiliations

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-3-0348-7698-8_15

DOI

http://dx.doi.org/10.1007/978-3-0348-7698-8_15

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1008400943


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